# Check if a number N starts with 1 in b-base

• Difficulty Level : Expert
• Last Updated : 23 Jun, 2022

Given a number N and base b if N in base b representation starts with 1 print Yes else print No

Examples :

```Input : n = 6, b = 4
Output : Yes
6 can be written as in base 4so answer is Yes as it starts with 1Input : n = 24, b = 2Output : Yes24 can be written as in base 2so answer is Yes as it starts with 1Input :  n = 24, b = 7Output : No24 can be written as in base 7so answer is No as it starts with 3        ```

When a number N is represented in base ‘b’ it gets converted to m+1 length sequence [Tex]b_{m-1}      [/Tex]….. which implies *  * …..+ * = N

The smallest number in base b and starting with ‘1’ i.e. 100..00 and m+1 digits in base is and largest number is 2* -1.So N should lie in this range. <= N <= 2* -1
Now another thing to notice is that m cannot exceed floor( (N)) because when we represent any number in base-2 it gets converted into a sequence of only 1s and 0s so the length of this sequence will always be greater than any other base representation and its length will be equal to floor( (N))+1.

So to check for a given base ‘b’ if N starts with 1 or not we will traverse from m = 1 to m = floor( (N)) and check if for any m N lies in the range <= N <= 2* -1 or not and accordingly print “Yes” or “No”.

## C++

 `// C++ program to check if number starts with``// one in base b representation``#include ``using` `namespace` `std;` `// Returns true if n starts with  1 in``// base b representation``bool` `CheckIfstartsWithOne(``int` `n, ``int` `b)``{``    ``// highest m can be log2(n)``    ``int` `m = log2(n);``    ``for` `(``int` `i = 1; i <= m; i++) {``        ``// if b^m <= N <= 2*b^m - 1,``        ``// return true``        ``if` `(n >= ``pow``(b, i) && n <= 2 * ``pow``(b, i) - 1)``            ``return` `true``;``    ``}``    ``return` `false``;``}` `// printing yes or no``void` `printYesORno(``int` `n, ``int` `b)``{``    ``if` `(CheckIfstartsWithOne(n, b) == ``true``)``        ``cout << ``"Yes"` `<< endl;``    ``else` `if` `(CheckIfstartsWithOne(n, b) == ``false``)``        ``cout << ``"No"` `<< endl;``}` `// driver function``int` `main()``{``    ``printYesORno(6, 4);``    ``printYesORno(24, 2);``    ``printYesORno(24, 7);``    ``printYesORno(24, 15);``}`

## Java

 `// Java program to check if number starts with``// one in base b representation` `class` `GFG {``    ` `    ``// Returns true if n starts with 1 in``    ``// base b representation``    ``static` `boolean` `CheckIfstartsWithOne(``int` `n, ``int` `b)``    ``{``        ` `        ``// highest m can be log2(n)``        ``int` `m = (``int``)(Math.log10(n) / Math.log10(``2``));``        ``for` `(``int` `i = ``1``; i <= m; i++) {``            ` `            ``// if b^m <= N <= 2*b^m - 1,``            ``// return true``            ``if` `(n >= (``int``)Math.pow(b, i) &&``                ``n <= ``2` `* (``int``)Math.pow(b, i) - ``1``)``                ``return` `true``;``        ``}``        ` `        ``return` `false``;``    ``}` `    ``// Driver method``    ``public` `static` `void` `main(String args[])``    ``{``        ``System.out.println(``           ``CheckIfstartsWithOne(``6``, ``4``) ? ``"Yes"` `: ``"No"``);``           ` `        ``System.out.println(``           ``CheckIfstartsWithOne(``24``, ``2``) ? ``"Yes"` `: ``"No"``);``           ` `        ``System.out.println(``           ``CheckIfstartsWithOne(``24``, ``7``) ? ``"Yes"` `: ``"No"``);``           ` `        ``System.out.println(``           ``CheckIfstartsWithOne(``24``, ``15``) ? ``"Yes"` `: ``"No"``);``    ``}``}`

## Python3

 `# Python3 program to check``# if number starts with one``# in base b representation``import` `math` `# Returns true if n``# starts with 1 in``# base b representation``def` `CheckIfstartsWithOne(n, b):``    ` `    ``# highest m can be log2(n)``    ``m ``=` `(``int``)(math.log2(n));``    ``for` `i ``in` `range` `(``1``, m ``+` `1``):``        ` `        ``# if b^m <= N <= 2*b^m - 1,``        ``#return true``        ``x ``=` `(``int``)(math.``pow``(b, i));``        ``if` `n >``=` `x ``and` `n <``=` `2` `*` `x ``-` `1``:``            ``return` `1``;``    ``return` `0``;` `# printing yes or no``def` `printYesORno(n, b):``    ``if` `CheckIfstartsWithOne(n, b) ``=``=` `1``:``        ``print``(``"Yes"``);``    ``if` `CheckIfstartsWithOne(n, b) ``=``=` `0``:``        ``print``(``"No"``);``        ` `# Driver Code``printYesORno(``6``, ``4``);``printYesORno(``24``, ``2``);``printYesORno(``24``, ``7``);``printYesORno(``24``, ``15``);` `# This code is contributed by mits.`

## C#

 `// C# program to check if number starts with``// one in base b representation``using` `System;` `class` `GFG{``    ` `// Returns true if n starts with 1 in``// base b representation``static` `bool` `CheckIfstartsWithOne(``int` `n, ``int` `b)``{``    ` `    ``// highest m can be log2(n)``    ``int` `m = (``int``)(Math.Log10(n) / Math.Log10(2));``    ``for``(``int` `i = 1; i <= m; i++)``    ``{``        ` `        ``// if b^m <= N <= 2*b^m - 1,``        ``// return true``        ``if` `(n >= (``int``)Math.Pow(b, i) &&``            ``n <= 2 * (``int``)Math.Pow(b, i) - 1)``            ``return` `true``;``    ``}``    ` `    ``return` `false``;``}` `// Driver code``public` `static` `void` `Main(String []args)``{``    ``Console.WriteLine(``       ``CheckIfstartsWithOne(6, 4) ? ``"Yes"` `: ``"No"``);``       ` `    ``Console.WriteLine(``       ``CheckIfstartsWithOne(24, 2) ? ``"Yes"` `: ``"No"``);``       ` `    ``Console.WriteLine(``       ``CheckIfstartsWithOne(24, 7) ? ``"Yes"` `: ``"No"``);``       ` `    ``Console.WriteLine(``       ``CheckIfstartsWithOne(24, 15) ? ``"Yes"` `: ``"No"``);``}``}` `// This code is contributed by Princi Singh`

## PHP

 `= pow(``\$b``, ``\$i``) &&``                ``\$n` `<= 2 * pow(``\$b``, ``\$i``) - 1)``            ``return` `true;``    ``}``    ``return` `false;``}` `// printing yes or no``function` `printYesORno(``\$n``, ``\$b``)``{``    ``if` `(CheckIfstartsWithOne(``\$n``, ``\$b``) == true)``        ``echo` `"Yes"` `,``"\n"``;``    ``else` `if` `(CheckIfstartsWithOne(``\$n``, ``\$b``) == false)``        ``echo` `"No"` `,``"\n"``;``}` `// Driver Code``printYesORno(6, 4);``printYesORno(24, 2);``printYesORno(24, 7);``printYesORno(24, 15);` `// This code is contributed by ajit``?>`

## Javascript

 ``

Output :

```Yes
Yes
No
Yes```

Time Complexity: O(m), where m is calculated as log2(n)
Auxiliary Space: O(1), as no extra space is required
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